TY - JOUR T1 - Validation of a stereological method for estimating particle size and density from 2D projections with high accuracy JF - bioRxiv DO - 10.1101/2022.10.21.513285 SP - 2022.10.21.513285 AU - Jason Seth Rothman AU - Carolina Borges-Merjane AU - Noemi Holderith AU - Peter Jonas AU - R. Angus Silver Y1 - 2022/01/01 UR - http://biorxiv.org/content/early/2022/10/25/2022.10.21.513285.abstract N2 - Stereological methods for estimating the 3D particle size and density from 2D projections are essential to many research fields. These methods are, however, prone to errors arising from undetected particle profiles due to sectioning and limited resolution, known as ‘lost caps’. A potential solution by Keiding et al. (1972) accounts for lost caps by quantifying the smallest detectable profiles in terms of their limiting section angle (ϕ). However, this simple solution has not been widely adopted nor validated. Rather, model-independent design-based stereological methods, which do not explicitly account for lost caps, have come to the fore. Here, we provide the first experimental validation of the Keiding model by quantifying ϕ of synaptic vesicles using electron-tomography 3D reconstructions. This analysis reveals a Gaussian distribution for ϕ rather than a single value. Nevertheless, curve fits of the Keiding model to the 2D diameter distribution accurately estimate the mean ϕ and 3D diameter distribution. While systematic testing using Monte Carlo simulations revealed an upper limit to determining ϕ, our analysis shows that mean ϕ can be used to estimate the 3D particle density from the 2D density under a wide range of conditions, and this method is potentially more accurate than minimum-size-based lost-cap corrections and disector methods. We applied the Keiding model to estimate the size and density of somata, nuclei and vesicles in rodent cerebella, where high packing density can be problematic for design-based methods.Competing Interest StatementThe authors have declared no competing interest.Tthickness of tissue section (transmission microscopy) or focal plane (ρz)D3D diameter of a particleμD ± σDmean and standard deviation of 3D particle diametersCVDσD / μDu.d.unit diameter, length normalised to μD (e.g. T/μD)planarT < 0.1 u.d.thinT ≈ 0.3 u.d.thickT ≥ 1 u.d.dobserved 2D diameter of a particledminminimum 2D diameter of a sample of particles47hminminimum penetration depth of a sample of particles42δminminimum 2D diameter of a given particle (z-stack analysis)dareaequivalent-area 2D diameter: darea = 2(area/π)½dshort, dlongshort and long-axis 2D diameterdgeometric(dshort·dlong)½davg½(dshort + dlong)μd ± σdmean and standard deviation of 2D diametersF(d)probability density of 3D diametersG(d)probability density of 2D diameters L(d) probability density of lost capsθparticle cap angle from section surface: sinθ = d/D where 0 ≤ θ ≤ 90θminequivalent cap angle of dmin: sinθmin = dmin/Dϕlower limit of θ where 0 ≤ ϕ ≤ 90ϕcutoffupper cutoff limit of when ϕ is determinableμϕ ± σϕmean and standard deviation of ϕCVϕσϕ / μϕdϕ:equivalent 2D diameter of ϕ dϕ = μD·sinϕζsection z-depth over which particle center points are sampledAreaxyROI xy-area over which particles are countedVFParticle volume fraction within a volume of interestAFParticle area fraction within a ROIN3DParticle count within a volume of interestN2DParticle count within a ROIλ3D3D particle density, λ3D = N3D / Volumexyzλ2D2D particle density, λ2D = N2D / AreaxyΩsum of projection overlaps for a given particle where Ω ≥ 0ψupper limit of Ω, i.e. 0 ≤ Ω ≤ ψχ2sum of squared differences between data and fits (or simulations)ΔParameter estimation error: % difference or difference from true valueμΔ ± σΔbias and (68%) confidence interval of a parameter’s estimation errorρxyzMicroscope resolutionRxyzImage/z-stack resolution ER -